Gradient flows and Ricci curvature for finite Markov chains

(Univ. Bonn)

Date: Feb 15, 2012,
time: 16:30

Place: Aula Dini (SNS)

Abstract. Since the seminal work of Jordan, Kinderlehrer and Otto, it is knownthat the heat flow on $R^n$ can be regarded as the gradient flow ofthe entropy in the Wasserstein space of probability measures.Meanwhile this interpretation has been extended to very generalclasses of metric measure spaces, but it seems to break down if theunderlying space is discrete.In this talk we shall present a new metric on the space of probabilitymeasures on a discrete space, based on a discrete Benamou-Brenierformula. This metric defines a Riemannian structure on the space ofprobability measures and it allows to prove a discrete version of theJKO-theorem.This naturally leads to a notion of Ricci curvature based on convexityof the entropy in the spirit of Lott-Sturm-Villani. We shall discusshow this is related to functional inequalities and present discreteanalogues of results from Bakry-Emery and Otto-Villani.This is partly joint work with Matthias Erbar (Bonn).